Talk:Mathematical logic

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Curious, since mathematics already uses logic, what part of logic does mathematics contribute which the art (not science)of logic does not already contain?

The history of logic distinguishes reasoning systems from logic, so math might contain a reasoning system independent of logic, but it is not logic! Therefore there is no such thing as mathematical logic. This would be a redundancy, not a tautology.

There are not dozens and dozens of types of logic; Analytic philosophers and mathematicians display an annoying disregard for English grammar and vocabulary, lazily preferring to designate any reasoning system as 'logic', commiting the heresy of dumbing down like any pedestrian.

I have yet to find a nexus between logic and math, so in my humble opinion, there is no overlap between the two.

Mathematical logic, also known as symbolic logic, is a separate field of mathematics, one that analyzes the systems of reasoning behind mathematics using symbols. Like Chalst said below, it is basically a synonym for formal logic. Lebob 00:53, 23 Dec 2004 (UTC)

I would hesitate to call Mathematical Logic a subfield of Mathematics. The logic of George Boole and C.S. Peirce may certainly be considered to be a subfield of mathematics as it is, in essence, Algebra limited to the number 1 and 0 (true and false). Other forms of mathematical logic such as that of Russell or Frege decreed that mathematics could be reduced to logic and not vice versa. So it is a bit of a grey area and maybe not something which should be stated in the first line of the article...? Aindriú Conroy 14:54 01 August 2006

Mathematical logic is generally considered a synonym for formal logic. Formal logic is taught mostly as a part of mathematics, which may answer part of your first question. If you could provide an example from the history of logic, where a reasoning system is distinguished from logic, that the second paragraph may make sense and may have some credibility. Unequivocally there are many types of logics, many different deductive systems, but the main ones used are sentential (sometimes called misleadingly propositional) or predicate logic, with minor differences (like how they treat identity operators). Predicate logic with identity is used in math courses as the basis for axiomatic set theory, for either Zermelo-Frankel or Godel-Von Neumann-Bernays variants, and no more logic is required than that. It would seem from an axiomatic point of view, mathematics is entirely dependent on and presupposes logic, however just like people learn how to speak before they ever take language courses, or play games before learning the rules, people can reason and count, before formally learning either logic or math. Logic is a very useful tool when doing mathematics, indispensible for proving mathematical theorems, doing any interesting work in mathematics. It would seem that set theory is indispensible to model theory, or the semantic part of formal logic. This leads to a circular situation, if you use model theory to prove theorems about logic, and logic to prove set theory theorems, and then set theory to prove model theoretic theorems, where does that leave you?

[edit] Questions and Quibbles

I've got quite a few complaints, and a couple of questions, about this article. I'll start here in the Talk page before upsetting anyone in my edits on the main page.

Although the layperson may think that mathematical logic is the logic of mathematics, the truth is rather that it more closely resembles the mathematics of logic

Mathematical logic refers to both, and my impression is that overall, in scientific contexts more commonly the former. See, eg. the FOM list.

Mathematical logic was the name given by Peano to what is also known as symbolic logic. In essentials, it is still the logic of Aristotle, but from the point of view of notation it is written as a branch of abstract algebra.

I haven't heard that the term "mathematical logic" is due to Peano. Do we have a source for this? (I'm not skeptical, just interested)
The stuff Peano was interested in was FOL, which certainly was not the logic of Aristotle;
Algebraic logic specifies a particular approach to mathematical logic that definitely is not the whole subject. There are also combinatorial, geometrical and computational approaches to logic, all of which need not be particularly algebraic. This last clause is misleading.

It was George Boole and then Augustus De Morgan, in the middle of the nineteenth century, who presented a systematic mathematical (of course non-quantitative) way of regarding logic.

Boole's contribution is regarded as brilliantly original, but also widely considered a mess, and De Morgan's attempts to clean it up don't really sort out the mess. Boole did originate the idea of treating logic algebraically, but to call him systematic is not right, IMO, and he doesn't seem to have been a big influence on the real originators of mathematical logic, Frege, Peano, Russell, et al.

Some changes proposed:

Much material missing from the History section, eg. invention of quantifier and function--argument notation, set theory, unmentioned pioneers (Frege, Cantor, Dedekind, Zermelo, Brouwer, Hilbert, Bernays, Weyl, Ackermann). I'd suggest settling on the time between Frege's Begriffschrift and the publication of Hilbert&Ackermann's work on FOL as the time when the foundations of modern mathematical logic were lain, earlier work is sort of "prehistory", later work, eg. Goedel, Gentzen, Tarski, Turing, is modern.
Rework "Topics in Logic": separate out applications of ML from the core subject.
Fundamental results is a nice idea, but it needs better structure. Maybe a timeline is the best idea? If not, by topic might work. The main problem is the huge difference in length of entries, but I think if we had a timeline in the history section, we could expand on them in a "core concepts" section.

Comments? ---- Charles Stewart 21:41, 25 Aug 2004 (UTC)

Mathematical logic has a link to symbolic logic, which redirects to "Mathematical logic". Change the link or create a new page for symbolic logic? Lebob 00:49, 23 Dec 2004 (UTC)

[edit] Technical reference

I would like to move the technical reference somewhere else because:

This page should be a general introduction to mathematical logic. It should be able to become a featured article. That means it should be accessible to people who don't have the technical background to read the technical reference.
While first-order logic is central to the field, it is not the only formal logic that is considered. Putting a technical reference here makes it seem like all mathematical logic uses first order logic.

I propose moving the technical reference section to its own page. CMummert 14:46, 14 July 2006 (UTC)

I moved it to Talk:First-order logic/Technical reference for safe keeping. CMummert 12:21, 29 October 2006 (UTC)

[edit] Comments and to-do list

This is a High importance article, but it is only at Start class. I read through it and noted the following places where it could be improved. I am noting them here so that I can work on them and others can comment on them. I assume in these comments that this article is, like Geometry, aimed at a reader who may have very little formal training in mathematics; thus formal statements of theorems will have to move elsewhere, and (correct and verifiable) intuition is the key here.

The lead and history sections have the same issues that Charles Stewart's comment from 2004 (above) discussed.
The section Fields of mathematical logic needs significant expansion. Each of proof theory, model theory, recursion theory, and set theory could have its own short para. The part on the MSC is probably not interesting to a general reader.
The relationship between mathematical logic and category theory is probably of interest to many readers.
A section on Foundations of mathematics is promised by the intro, and needs to be added.
The Fundamental results section is too terse for an untrained reader.
The See also list is way too long. Most of those links could be integrated into the article body. CMummert 12:28, 29 October 2006 (UTC)